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        CK + \[Gamma]G*GK;\), "\[IndentingNewLine]", 
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      StyleBox[\( (*\ Then\ guess\ \(\(solution\)\(:\)\)\ \ *) \),
        FontFamily->"Times New Roman",
        FontWeight->"Plain",
        FontSlant->"Plain",
        FontTracking->"Plain",
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        FontWeight->"Plain",
        FontSlant->"Plain",
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      StyleBox[\( (*\ \ k'\  = \ \[Eta]*k\  + \ H*a\ \ *) \),
        FontFamily->"Times New Roman",
        FontWeight->"Plain",
        FontSlant->"Plain",
        FontTracking->"Plain",
        FontVariations->{"Underline"->False,
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        "StrikeThrough"->False,
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        "RotationAngle"->0}], "\[IndentingNewLine]", 
      StyleBox[\( (*\ \ c\  = \ Ac*k\  + \ Bc*a; \ \ n\  = \ 
            An*k\  + \ Bn*a; \ \ w\  = \ Aw*k\  + \ Bw*a; \ \ y\  = \ 
            Ay*k\  + \ By*a\ \ *) \),
        FontFamily->"Times New Roman",
        FontWeight->"Plain",
        FontSlant->"Plain",
        FontTracking->"Plain",
        FontVariations->{"Underline"->False,
        "Outline"->False,
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        "StrikeThrough"->False,
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        FontFamily->"Times New Roman",
        FontWeight->"Plain",
        FontSlant->"Plain",
        FontTracking->"Plain",
        FontVariations->{"Underline"->False,
        "Outline"->False,
        "Shadow"->False,
        "StrikeThrough"->False,
        "Masked"->False,
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      StyleBox[\( (*\ By\ using\ method\ of\ undetermined\ coefficnets, \ 
          easy\ to\ get\ set\ of\ equations\ below*) \),
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        FontWeight->"Plain",
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          0}]}], "\[IndentingNewLine]", \(eqa1 = \(-\[Eta]\)*\((\[CapitalXi]*\
\((1 - \[Alpha])\)*\((1 - An)\) + Z1*Ac)\) + 
          Z2*Ac;\), "\[IndentingNewLine]", \(eqb1 = \
\(-H\)*\((\[CapitalXi]*\((1 - \[Alpha])\)*\((1 - An)\) + Z1*Ac)\) - 
          Z1*Bc*\[Rho] + Z2*Bc + 
          Z3 + \[CapitalXi]*\((1 - \[Alpha])\)*
            Bn*\[Rho];\), "\[IndentingNewLine]", \(eqa2 = 
        Aw + Z4*Ac + \[Alpha] - \[Alpha]*
            An;\), "\[IndentingNewLine]", \(eqb2 = 
        Bw + Z4*Bc - \[Alpha]*Bn + Z5;\), "\[IndentingNewLine]", \(eqa3 = 
        Z6*Ac + \[Psi]*An - Aw;\), "\[IndentingNewLine]", \(eqb3 = 
        Z6*Bc + \[Psi]*Bn - Bw + Z7;\), "\[IndentingNewLine]", \(eqa4 = 
        Z8 + Z9*An + Z10*Ac - Ay;\), "\[IndentingNewLine]", \(eqb4 = 
        Z9*Bn + Z10*Bc + Z11 - 
          By;\), "\[IndentingNewLine]", \(eqa5 = \(-\[Eta]\) + 1 - \[Delta] + 
          YK*Ay - Z12*Ac;\), "\[IndentingNewLine]", \(eqb5 = \(-H\) + YK*By - 
          Z12*Bc - Z13;\)}], "Input"],

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Cell[BoxData[
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      StyleBox[\( (*\ 
          I\ solve\ the\ system\ in\ parts\ because\ it\ is\ numerically\ \
much\ faster . \ First, \ note\ that\ eqa1, eqa2, eqa3, eqa4, 
          eqa5\ depend\ only\ on\ Aw, An, Ay, 
          Ac, \[Eta] . \ Therefore\ we\ can\ solve\ this\ sub - 
            system\ of\ 5\ equations\ separately\ for\ these\ 5\ unknowns . \ 
                Moreover, \ 
          note\ that\ the\ last\ 4\ equations\ are\ linear\ in\ Aw, An, Ay, 
          Ac, \ so\ can\ be\ easily\ solved\ as\ a\ function\ of\ the\ \
unknowns*) \),
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        "Outline"->False,
        "Shadow"->False,
        "StrikeThrough"->False,
        "Masked"->False,
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      RowBox[{\(temp1 = 
            Simplify[
              Solve[{eqa2 \[Equal] 0, eqa3 \[Equal] 0, eqa4 \[Equal] 0, 
                  eqa5 \[Equal] 0}, {Aw, An, Ay, Ac}]];\), 
        "\[IndentingNewLine]", \(temp2 = temp1[\([1]\)];\), 
        "\[IndentingNewLine]", \(temp2 /. passive\), 
        "\[IndentingNewLine]", \(Aw = \(temp2[\([1]\)]\)[\([2]\)]; 
        An = \(temp2[\([2]\)]\)[\([2]\)]; Ay = \(temp2[\([3]\)]\)[\([2]\)]; 
        Ac = \(temp2[\([4]\)]\)[\([2]\)];\), 
        "\[IndentingNewLine]", \(temp3 = Solve[eqa1 \[Equal] 0, \[Eta]]; \ 
        temp3 /. passive\), 
        "\[IndentingNewLine]", \(\[Eta] = \
\(\(temp3[\([1]\)]\)[\([1]\)]\)[\([2]\)];\), 
        "\[IndentingNewLine]", \(\[Eta] /. passive\), 
        "\[IndentingNewLine]", \(ClearAll[temp1, temp2, temp3];\), 
        "\[IndentingNewLine]", 
        StyleBox[\( (*Now\ solve\ for\ the\ other\ 5\ components, \ H, Bc, 
            Bw, Bn, By, \ using\ the\ other\ 5\ equations, \ eqb1, eqb2, 
            eqb3, eqb4, \(\(eqb5\)\(.\)\)*) \),
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          "Shadow"->False,
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        "\[IndentingNewLine]", \(temp1 = 
            Solve[{eqb1 \[Equal] 0, eqb2 \[Equal] 0, eqb3 \[Equal] 0, 
                eqb4 \[Equal] 0, eqb5 \[Equal] 0}, {H, Bc, Bw, Bn, By}];\), 
        "\[IndentingNewLine]", \(temp2 = temp1[\([1]\)]; \ temp2 /. passive\),
         "\[IndentingNewLine]", 
        RowBox[{\(H = \(temp2[\([3]\)]\)[\([2]\)]\), 
          ";", \(Bc = \(temp2[\([4]\)]\)[\([2]\)]\), 
          ";", \(Bw = \(temp2[\([1]\)]\)[\([2]\)]\), 
          ";", \(Bn = \(temp2[\([2]\)]\)[\([2]\)]\), 
          ";", \(By = \(temp2[\([5]\)]\)[\([2]\)]\), ";", 
          "\[IndentingNewLine]", 
          
          StyleBox[\( (*Below\ are\ displayed\ the\ solutions\ for\ each\ \
component, \ evaluated\ at\ the\ passive - rule\ coefficients*) \),
            FontFamily->"Times New Roman",
            FontWeight->"Plain",
            FontSlant->"Plain",
            FontTracking->"Plain",
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        0.6466890145066807`\ \((\(-1.2343315172974811`\) + 1.`\ \[Eta])\), 
      Ay \[Rule] \(-2.0669897463378453`\) + 2.119989483423431`\ \[Eta] - 
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                3.66`\ \[Eta])\), 
      Ac \[Rule] 
        0.6466890145066807`\ \
\((\(\(3.9918983267457184`\)\(\[InvisibleSpace]\)\) - 
              3.66`\ \[Eta])\)}\)], "Output"],

Cell[BoxData[
    \({{\[Eta] \[Rule] 0.75482850821617`}, {\[Eta] \[Rule] 
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      StyleBox[\( (*Next, \ 
          I\ solve\ for\ the\ MA\ coefficients\ and\ plot\ IRF\ in\ passive - 
            rule\ case\ just\ for\ \(\(check\)\(.\)\)*) \),
        FontFamily->"Times New Roman",
        FontWeight->"Plain",
        FontSlant->"Plain",
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        "Shadow"->False,
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      "\[IndentingNewLine]", \(ClearAll[passc, passy, plotc, 
        ploty]\[IndentingNewLine]
      \[Mu]c = Bc; \ \[Nu]c = Ac*H - Bc*\[Eta]; \[IndentingNewLine]\[Mu]n = 
        Bn; \ \[Nu]n = An*H - Bn*\[Eta]; \[IndentingNewLine]\[Mu]w = 
        Bw; \ \[Nu]w = Aw*H - Bw*\[Eta]; \[IndentingNewLine]\[Mu]y = 
        By; \ \[Nu]y = 
        Ay*H - By*\[Eta]; \[IndentingNewLine]{\[Mu]c, \[Nu]c, \[Mu]n, \ \
\[Nu]n, \[Mu]w, \ \[Nu]w, \[Mu]y, \ \[Nu]y} /. passive\[IndentingNewLine]
      passy[1] = \[Mu]y /. passive; 
      passy[2] = \((\((\[Eta] + \[Rho])\)*passy[1] + \[Nu]y)\) /. 
          passive; \[IndentingNewLine]\(Do[
          passy[i] = \((\[Eta] + \[Rho])\)*passy[i - 1] - \[Eta]*\[Rho]*
                  passy[i - 2] /. passive, {i, 3, 16}];\)\[IndentingNewLine]
      passc[1] = \[Mu]c /. passive; 
      passc[2] = \((\[Eta] + \[Rho])\)*passc[1] + \[Nu]c /. 
          passive; \[IndentingNewLine]\(Do[
          passc[i] = \((\[Eta] + \[Rho])\)*passc[i - 1] - \[Eta]*\[Rho]*
                  passc[i - 2] /. passive, {i, 3, 16}];\)\[IndentingNewLine]
      ploty = Array[passy, 16]; 
      ListPlot[ploty, PlotJoined \[Rule] True, 
        PlotLabel \[Rule] "\<IRF of output under passive policy\>"]\
\[IndentingNewLine]
      plotc = Array[passc, 16]; 
      ListPlot[plotc, PlotJoined \[Rule] True, 
        PlotLabel \[Rule] "\<IRF of consumption under passive policy\>"]\
\[IndentingNewLine]
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